1. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Daniel F. V. JAMES Department of Physics & Center for Quantum Information and Quantum Control University of Toronto QELS ’10, San Jose CA QFF-Quantum State Reconstruction/QFF-1 21 May 2010 Measuring and Characterizing Quantum States and Processes

2. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Outline 1. Single qubit tomography = Polarimetry 3. Positivity: The first big problem, and its fixes. 2. Two (and more) photons (and other qubits). 4. Characterizing quantum states. 5. Quantum Processes. 6. Scalability: the second big problem. 7. Conclusions and resources.

3. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 §1: State of a Single Qubit Photon polarization based qubits Measure multiple (assumed identical) copies: frequency of “clicks” gives estimate of |  | 2 Measure Single Copy by projecting on to : get answer “click” or “no click” –One bit of information about  and  : you know one of them is non-zero

4. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Find relative phase of  and  by performing a unitary operation before beam splitter: e.g.: Frequency of “clicks” now gives an estimate of Systematic way of getting all the data needed….

5. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Stokes Parameters G. G. Stokes, Trans Cambridge Philos Soc 9 399 (1852) Measure intensity with four different filters: (iv) RCP (i) 50% intensity (iii) 45 o polarizer (ii) H-polarizer

6. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Stokes Parameters These 4 parameters completely specify polarization of beam Beam is an ensemble of photons… Pauli matrices

7. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Outline 1. Single qubit tomography = Polarimetry 2. Two (and more) photons (and other qubits).

8. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Pure states –Ideal case Mixed states –Quantum state is random: need averages and correlations of coefficients §2: Two Qubit Quantum States

9. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 State Creation by OPDC Arbitrary state: change basis…

10. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Coincidence Rate measurements for two photons Two-Qubit Quantum State Tomography source measurement

11. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Doesn’t give the right answer…. Linear combination of n a,b yields the two-photon Stokes parameters: From the two-photon Stokes parameters, we can get an estimate of the density matrix:

12. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Outline 1. Single qubit tomography = Polarimetry 3. Positivity: The first big problem, and its fixes. 2. Two (and more) photons (and other qubits).

13. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 §3: The First Problem… Properties of Density matrices: - Hermitian: - unit trace: - non-negative definite  all eigenvalues are non- negative: What about: ?  2/3 - must try harder! “happy beach” of positive states “sea” of negative matrices experimental data (with error bars) Q: how to find the “best” positive  from noisy data?

14. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Goodness? Data is random, with (say) a Gaussian distribution, with mean given by the expectation values determined by the density matrix: find the “best”  by ensuring this probability is a maximum

15. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Enforcing Positivity We need a  such that Incorporate constraints in numerical search so that the eigenvalues are positive -tedious mucking about finding the eigenvalues each step -a bunch of Lagrange multipliers to find “Cholesky decomposition” - André-Louis Cholesky (French Army Officer, 1875-1918) - non-negative matrices can be written as follows:

16. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Maximum Likelihood fit to "physical" density matrix –Density matrix must be Hermitian, normalized, non-negative –Numerically Minimize the function: Maximum Likelihood Tomography* –where:  = TT † /Tr{TT † } and * D. F. V. James, et al., Phys Rev A 64, 052312 (2001).

17. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Example: Measured Density Matrix

18. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Quantum State Tomography I Sublevels of Hydrogen (partial) (Ashburn et al, 1990) Optical mode (Raymer et al., 1993) Molecular vibrations (Walmsley et al, 1995) Motion of trapped ion (Wineland et al., 1996) Motion of trapped atom (Mlynek et al., 1997) Liquid state NMR (Chaung et al, 1998) Entangled Photons (Kwiat et al, 1999) Entangled ions (Blatt et al., 2002; 8 ions: 2005) Superconducting qubits (Martinis et al., 2006)

19. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Outline 1. Single qubit tomography = Polarimetry 3. Positivity: The first big problem, and its fixes. 2. Two (and more) photons (and other qubits). 4. Characterizing quantum states.

20. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 §4: Characterizing the State Purity Fidelity: how close are two states? Pure states: Mixed states: doesn’t work:

21. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Pure states How much entanglement is in this state? Measures of Entanglement –Concurrence: –Concurrence is equivalent to Entanglement : –C=0 implies separable state –C=1 implies maximally entangled state (e.g. Bell states) –Entropy of reduced density matrix of one photon

22. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Mixed states can be de-composed into incoherent sums of pure (non-orthogonal) states: Entanglement in Mixed States “Average” Concurrence: dependent on decomposition “Minimized Average Concurrence”: –Independent of decomposition –C=0 implies separable state –C=1 implies maximally entangled state (e.g. Bell states) –Analytic expression (Wootters, ‘98) makes things very convenient!

23. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Two Qubit Mixed State Concurrence Transpose (in computational basis) “spin flip matrix” Eigenvalues of R (in decreasing order) W.K. Wootters, Phys. Rev. Lett. 80, 2245 (1998)

24. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 “Map” of Hilbert Space* * D.F.V. James and P.G.Kwiat, Los Alamos Science, 2002 MEMS states *W. J. Munro et al., Phys Rev A 64, 030302-1 (2001)

25. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Outline 1. Single qubit tomography = Polarimetry 3. Positivity: The first big problem, and its fixes. 2. Two (and more) photons (and other qubits). 4. Characterizing quantum states. 5. Quantum Processes.

26. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 §5: Process Tomography Trace Preserving Completely Positive Maps: Every thing that could possibly happen to a quantum state “operator-sum formalism” “Kraus operators” set of basis matrices, e.g.: Trace orthogonality:

27. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Decompose the Kraus operators:  is a Hermitian, positive 16x16 matrix(“error correlation matrix”), with the constraints- where- then-  is almost like  “Choi-Jamiolkowski isomorphism”

28. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Estimate probability from counts 16x16 = 256 data: Recover   by linear inversion - problematic in constraining positivity - close analogy with state tomography… Process Tomography 16 Input states 16 Projection states *I. L. Chuang and M. A. Nielsen, J. Mod Op. 44, 2455 (1997) *

29. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Numerically optimize where: (256 free parameters) Constraints on   : - positive - Hermitian - additional constraint for physically allowed process: Maximum Likelihood Process Tomography

30. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Process Tomography of UQ Optical CNOT* Actual CNOT Most Likely   matrix *J. L. O’Brien et al., “Quantum process tomography of a controlled-NOT gate,” Phys Rev Lett, 93, 080502 (2004); quant-ph/0402166.

31. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Outline 1. Single qubit tomography = Polarimetry 3. Positivity: The first big problem, and its fixes. 2. Two (and more) photons (and other qubits). 4. Characterizing quantum states. 5. Quantum Processes. 6. Scalability: the second big problem.

32. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 §6: Scalability? record: 8 qubit W-state (Blatt et al., 2006) Why not more? N qubit state tomography requires 4 N -1 measurements (& numerical optimization in a 4 N -1 dimensional space)

33. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Fixes? Measurements: you can get a good guess at the density matrix with fewer measurements (it still requires exponential searching) (Aaronson, 2006) Direct Characterization: In some cases you can get the information you need more directly, without the tedious mucking around with the density matrix (e.g. entanglement witnesses; noise characterization) Push the envelope: How far can we go using smart computer science before we hit the wall? - convex optimization - improved data handling and processing -other approaches to ‘optimziation’

34. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 - ‘quick and dirty’: zero out the negative eigenvalues rather than perform an exhaustive optimization. - ‘forced purity’: we are trying to make specifc states, so why not use that fact? Are we being too pedantic in looking for the optimal density matrix to fit a given data set, when a simpler numerical technique produces a good estimation (i.e. within the error bars)? Is “the best” the enemy of “good enough” Alternatives: Both give positive matrices quickly: but are they the actual states in question? M. Kaznady and D. F. V. James, Phys Rev A 79, 022109 (2009); arXiv:0809.2376.

35. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 - Fidelities for 2-qubit states: Numerical experiments - choose a state - simulate measurement data with a Poisson RNG - estimate state using code -compare estimated and actual state MLE Q&D FP

36. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Bayesian Approach* *Robin Blume-Kohout, “Optimal,reliable estimation of quantum states,” New Journal of Physics 12 043034 (2010)

37. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Polynomial Time Tomography?* *S. T. Flammia et al., “Heralded Polynomial-Time Quantum State Tomography” arXiv:1002:3839; see also M. Cramer and M. Plenio, arXiv:1002.3780 1.Choose a few-parameter set of states suitable for what you are trying to do. 2.Find a protocol to find the best set of parameters to fit your data. 3.Check that the state thus recovered is a good approximation to the actual data

38. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Outline 1. Single qubit tomography = Polarimetry 3. Positivity: The first big problem, and its fixes. 2. Two (and more) photons (and other qubits). 4. Characterizing quantum states. 5. Quantum Processes. 6. Scalability: the second big problem. 7. Conclusions and resources.

39. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Conclusions Maybe these techniques can give reasonably good characterization of a dozen or so qubits…. Beyond that, how an we know quantum computers is doing what it’s meant to? - well characterized components. - error correction: you can’t know if it’s bust or not, so you’d best fix it anyway. - answers are easy to check.

40. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Resources D.F.V. James and P.G.Kwiat, “Quantum State Entanglement: Creation, characterization, and application,” Los Alamos Science, 2002 A gentle introduction: More details (books, compilation volumes etc.): U. Leonhardt, Measuring the Quantum State of Light (Cambridge, 1997) [tomography of harmonic oscillator modes via inverse Radon transforms] Quantum State Estimation, Lecture Notes in Physics, Vol. 649, M. G. A. Paris and J. Řeháček, eds. (Springer, Heidelberg, 2004) Asymptotic Theory of Quantum Statistical Inference: Selected Papers, edited by M. Hayashi (World Scientific, Singapore, 2005)

41. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Thanks to... Dr. René Stock Asma Al-Qasimi Omal Gamel Max Kaznady My Group: Funding Agencies: Ardavan Darabi Faiyaz Hasan Timur Rvachov Bassam Helou Collaborators/helpers: Paul Kwait Andrew White Bill Munro Christian Roos Hartmut Häffner Steve Flammia Robin Blume- Kohout